Advanced Numerical Methods for Solving Differential Equations with Applications in Science and Engineering

Dear Colleagues,

Numerical integration methods are an essential tool for implementing continuous systems described by differential equations in digital computers.

The developments in this field of mathematics have resulted in several large classes of so-called differential equation solvers, including single- and multistep algorithms with fixed and adaptive timesteps, as well as various space discretization methods, e.g., finite element analysis. In some areas of science and engineering, numerical simulation is the only way of obtaining reliable solutions. The choice of the numerical method and the simulation parameters for a certain problem is still a challenging task, which is commonly solved empirically. The time-efficient and reliable solution of stiff and/or nonlinear non-stationary equations with the required accuracy is a crucial goal for scholars in the field. One of the key branches in numerical methods is special solvers for nonlinear, e.g., chaotic systems, which are highly sensitive to all simulation parameters, including discretization operator type. A better understanding of the impact that numerical methods have on the properties of finite-difference models of continuous nonlinear systems could allow researchers to reveal the limits of computer simulation and predictability of nonlinear dynamics.

This Special Issue is dedicated to advanced numerical integration techniques, including single- and multistep methods, collocation methods, and composition and extrapolation ODE and PDE solvers. New finite-difference schemes for solving PDEs, adaptive stepsize control methods, parallelization on GPUs, and various applications of numerical simulation are also of interest.

Papers may report on original research in numerical integration, discuss methodological aspects, review the current state of the art, or offer perspectives on future prospects.

Specific methods and fields of applications include, but are not limited to:

• Symplectic and symmetric integrators;
• Hardware-targeted methods;
• Multistep ODE solvers;
• Semi-implicit integration and composition methods;
• Nonlinear integration methods;
• Step-control techniques and truncation error estimation;
• Special techniques for solving nonlinear systems;
• Numerical methods of solving time-dependent PDEs;
• Large-scale and multiscale PDEs;
• PDEs in spatially inhomogeneous media.

Papers on fractional, perturbed, or delay differential equations are welcomed only if they clearly show a direct application in science or engineering.

Manuscripts using only traditional methods are of interest only if they conduct an extensive comparison of the performance of different methods.   

Dr. Endre Kovács
Dr. Denis Butusov
Dr. Valerii Ostrovskii
Guest Editors

Manuscript Submission Information

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